Triangle of Powers: Revolutionary Math Notation

[YoungPhysicist]

Recently I saw this clip on YouTube which has a revolutionary idea of mathematical notations,What do you think about it?

 

[jedishrfu]

@Young physicist thanks for sharing this clip.

The change of notation reminded me of some things I've learned in the past:

APL was a notational programming language that almost took off as a computing specification language. IBM used it to describe their opcodes. Economists used it for their models and its still here today but in modified form since keyboards don’t typically support Greek letters and overstrikes. Programmers joked that it was a write only language unreadable after 6 months from writing it.

APL was developed by Iverson and he was trying to marry mathematical notation to computing. APL was a multiple array programming language which used greek letters and overstrike letters as operators in the language. However, it was distinctly different from traditional math notation and ultimately changed back into more of a programming language since most keyboard don't have greek letters and don't do overstrikes as needed by APL.

https://en.wikipedia.org/wiki/APL_(programming_language)

Mathematical notation[edit]
A mathematical notation for manipulating arrays was developed by Kenneth E. Iverson, starting in 1957 at Harvard University. In 1960, he began work for IBM where he developed this notation with Adin Falkoff and published it in his book A Programming Language in 1962.[2] The preface states its premise:

Applied mathematics is largely concerned with the design and analysis of explicit procedures for calculating the exact or approximate values of various functions. Such explicit procedures are called algorithms or programs. Because an effective notation for the description of programs exhibits considerable syntactic structure, it is called a programming language.

This notation was used inside IBM for short research reports on computer systems, such as the Burroughs B5000 and its stack mechanism when stack machines versus register machines were being evaluated by IBM for upcoming computers.

...

As early as 1962, the first attempt to use the notation to describe a complete computer system happened after Falkoff discussed with William C. Carter his work to standardize the instruction set for the machines that later became the IBM System/360 family.

...

One of the motivations for this focus of implementation was the interest of John L. Lawrence who had new duties with Science Research Associates, an educational company bought by IBM in 1964. Lawrence asked Iverson and his group to help use the language as a tool to develop and use computers in education.[12]

...

A key development in the ability to use APL effectively, before the wide use of cathode ray tube (CRT) terminals, was the development of a special IBM Selectric typewriter interchangeable typing element with all the special APL characters on it. This was used on paper printing terminal workstations using the Selectric typewriter and typing element mechanism, such as the IBM 1050 and IBM 2741 terminal.

...

Many APL symbols, even with the APL characters on the Selectric typing element, still had to be typed in by over-striking two extant element characters. An example is the grade up character, which had to be made from a delta (shift-H) and a Sheffer stroke (shift-M). This was necessary because the APL character set was much larger than the 88 characters allowed on the typing element, even when letters were restricted to upper-case (capitals).

 

[jedishrfu]

As I watched the video, I realized it would a stretch to have students actively use it but I did something like this in high school I had little diagrams for trig functions that I wrote at the top of my paper. In that way, I was able to breeze thru the test with the confidence that I wouldn't remember something wrong.

Our trigonometry teacher was a stickler on test time. He once said he takes his own tests and a 45 min test for us would take him 5 minutes. For the best of us, it took a full 45 mins and we were sweating at the end with achy fingers. Of course, we didn't believe his boast until he showed us the shortcuts that he used and would allow (mostly in needless graph labeling) but that we hadn't considered. So I learned a valuable lesson from him and began my tiny diagrams for math exploration.

I can see some students watching this video and getting confused but I can see others who will master it along with the traditional concepts driven by curiosity and the need to be unique in school. In my case, I have a kind of dyslexia where I would confuse left and right and so would have trouble with perfectly symmetrical things like this notation to remember it correctly. His initial visuals of finding the missing corner were great but when he started to slide things together and rotate the triangle about I began to feel that I wouldn't remember it correctly in a pinch.

Also, I can compare this new notation to our switch from roman numerals to decimal numbers, or the move from Maxwell's notation to vector analysis and later to differential forms with differential forms bringing out certain symmetries that you didn't see in Maxwell's work or in vector analysis. However I must caution that students can play with this notation but that they must be careful to understand the traditional notation. I think in playing with it, students will hone their math skills and insight even more but at the risk of getting thoroughly confused.

Another problem with this new notation is that its applying math concepts to a triangle analogy that you can manipulate with triangle rules. The problem with analogies is that students can take them too far into realms where they no longer work.

One famous example, is the rubber sheet with a ball in the center analogy to represent the sun illustrates the curvature of space. Students can get a lot of insight from it but some of that insight as you dig deeper is flawed.
- Imagining the rubber sheet in outer space and the ball wouldn't stretch the sheet what does that mean?
- or what does it mean that the sheet stretches downward in the Z direction?
- is there some new force in the Z direction here needed to describe the curvature?

As a student you wouldn't know when to stop and for relativity in particular these visuals cause a lot of headaches and confusion and so we also need to go back to the core math of relativity and spacetime diagrams which aren't as intuitive to understand what's really going on.

https://medium.com/the-physics-arxi...bber-sheet-is-not-like-spacetime-b8566ba5a110



Anyway, you can see how powerful this analogy is and yet there are dragons hidden within it.

One last point, I remember how a visual symmetry had helped me on a test for nth roots of a number in the complex plane. I couldn't remember the formula we were taught (confusing sin and cos in it ie which was the real part and which was the imaginary part) but did remember the geometric symmetry and was able to properly compute the roots from that symmetry saving the day for me on that test and my ultimate grade.

Bottom line is use the triangle of power to further your understanding and insight but don't become too attached to it instead look to gain knowledge from it and look to develop your own notation that will help you. Remember that traditional math notation has been honed over a few millenia to what we see today and where it works it does so astonishingly well and where it doesn't mathematicians will find better notation for future generations. Things just move generationally slow and the triangle of power is one small step that may or may not take off in the next generation.

 

[YoungPhysicist]

I don’t have all the experience and insight that you have since I am still a student but as far as I am concerned,although this notation an explain exponents,nth roots and logarithms,it doesn’t seem like a viable way to incorporate into a larger equation with other notations still remain the same, unless you are learning math the intuitive way from the beginning,which is not our generation.

 

[jedishrfu]

The takeaway here is that you as a student can be creative and develop memory techniques to gain more insight into things. I did it with my tiny diagrams and you can do it too. Sometimes, I think its my lack of a good memory or that I have a visual learning style that has helped me develop a sense of creativity. I use the skill to fill in what I can't remember.

Sometimes, it leads to new insights and sometimes to wrong ideas but its how we learn.

Also I think that's why I love the videos produced by 3brown1blue.

 

[YoungPhysicist]

3blue1brown

 

[jedishrfu]

3blue1brown

Dyselxia I mean Dyslexia.

When I wrote it, I was thinking blue is 1/4 and brown is 3/4 of the visual icon he uses. But I guess I interchanged the colors in my mind.

 

[YoungPhysicist]

Dyselxia I mean Dyslexia.

When I wrote it, I was thinking blue is 1/4 and brown is 3/4 of the visual icon he uses. But I guess I interchanged the colors in my mind.

Ha

 

[jedishrfu]

Another story regarding Ramanujan. He developed his own math notation and mathematicians of the day couldn't understand what he was working on and rejected his work as being a crackpot or charlatan. However, Hardy saw genius there and invited him to England. Ramanujan was an intuitive mathematician who just saw and understood things. Many times though they were wrong but a significant amount I think about a third was true and completely original.

There's the story of the taxicab numbers 1729 where Hardy went to visit him in the hospital and remarked that even the taxicab number was boring. Ramanujan replied oh no it is really quite interesting it is the smallest number that can be represented as the sum of two cubes two different ways 9^3 + 10^3 and 12^3 + 1^3



More recently, it was discovered in Ramanujan's notebooks that he was playing with a whole collection of these numbers and was in essence trying to find an example to disprove Fermat's little theorem.



https://plus.maths.org/content/ramanujan

Hardy's book on Ramanujan:

https://www.amazon.com/gp/product/0821820230/?tag=pfamazon01-20

Lastly, I'd like to mention a cool book that I've found the piques my amateur interest in math:

Math 1001 by Prof Elwes has a lot of short descriptions of mathematical things that you can on and research more:

https://www.amazon.com/dp/1554077192/?tag=pfamazon01-20

 

[OmCheeto]

I found the following comment from the video rather amusing; "Just think about how confusing logarithms were the first time that you learned about them."

The first time? Ha! I've been tinkering with maths for 45 years now, and still have to pull out a cheat sheet of logarithms and exponents every time I have to use them in a problem. I think I'll give these new triangles a try.

 

[fresh_42]

I found the following comment from the video rather amusing; "Just think about how confusing logarithms were the first time that you learned about them."

The first time? Ha! I've been tinkering with maths for 45 years now, and still have to pull out a cheat sheet of logarithms and exponents every time I have to use them in a problem. I think I'll give these new triangles a try.

If I see $\log_b\; , \; \sqrt[n]{.} \; , \;a^m$ I know immediately all allowed laws and rules. If someone gave me this, sorry incredible stupid triangular, all I have a chance with is to confuse which corner tells me what. And do we also omit the basis like we do if it's the Euler constant?

Switch $e$

Case "drop"

Unknowns and basis are indistinguishable.​

Case "write"

Additional notation and advantage is gone.
​

Switch $\mathbb{F}$

Case "$\mathbb{R}$"

Endless discussion with students and their handwritings.​

Case "$\mathbb{C}$"

Inappropriate use.​

Switch "confusion"

Case "$\Delta$"

Inappropriate use.​

Case "triangles"

Inappropriate use.​

And finally the definitions:
$$ {}_{e}\stackrel{x+y}{\bigtriangleup }_{}= {}_{e}\stackrel{x}{\bigtriangleup }_{} \cdot {}_{e}\stackrel{y}{\bigtriangleup }_{} \text{ and } {}_{e}\stackrel{}{\bigtriangleup }_{ab}= {}_{e}\stackrel{}{\bigtriangleup }_{a} + {}_{e}\stackrel{}{\bigtriangleup }_{b}$$

or even better:
$$ \stackrel{x+y}{\bigtriangleup }_{}= \stackrel{x}{\bigtriangleup }_{} \cdot \stackrel{y}{\bigtriangleup }_{} \text{ and } \stackrel{}{\bigtriangleup }_{ab}= \stackrel{}{\bigtriangleup }_{a} + \stackrel{}{\bigtriangleup }_{b}$$
No way I will ever not confuses this!

 

[OmCheeto]

Interestingly, I've found only one other article that discusses the subject beyond just reporting about its existence:

https://www.science4all.org/article/the-triangle-of-power/

I followed the authors suggestion, and worked through most of the identities that I use, translating them into this new notation.
I only had one that I couldn't reconcile: ln(x^y) = yln(x)

The author has a PhD in applied mathematics, and seems to enjoy the triangle.
And given that he has a PhD, and I know maths at the 3rd grade level, he started talking big maths words about halfway through, so I couldn't follow any of that.

Anyways, I suspect it will take me many more hours of practice to get even a basic handle on this.

 

[jedishrfu]

That’s a great supporting link. His discussion showed how much you have to change how you think to use it. The problem for students wanting to use this will be teacher resistance. This often takes a few generations if successful to become a mainstream notation.

Years ago I learned about the Trachtenberg multiplication scheme that could transform an ordinary student into a calculating wizard. It was quite beautiful, learning multiplication without tables but it never took off. I was fascinated by it because I struggled to learn those tables and would always flub 9x6 vs 8x7 is it 54 or 56. It wasn’t until I learned the nines rule that I was able straighten this out.

Another great idea that has yet to catch on is the hyperreals and how they eliminate the need for limits in calculus for deriving the rules of differentiation of a function. One mathematician even developed a whole Calculus book premised on it but it has yet to modernize Calculus courses. Imagine how simple it is to apply simple algebra and cross out dx factors, hyperreals allows this and maps to reals while placing it on a firm mathematical footing.

 

[symbolipoint]

Video made no impression for me. We should learn the symbolism and the notation directly from visual information. This is why students can learn basic and intermediate algebra (although such still requires effort). We are shown instructive pictures and these are directly transformed into symbolism and notations. Think back a few years! We first learned the smaller counting numbers through pictures AND real items; and then the corresponding symbols were shown, explained, and demonstrated.

small edit: appended to last sentence

 

[fresh_42]

I do not think it is a matter of getting used to, or overcome resistance, or simply time. It is a matter of meaningfulness.

• I most often write terms like $a^n$ within group theory. Unthinkable to change to ${}_a\stackrel{n}{\triangle}$. It is simply as unnecessary as it is impracticable.
• The current notation is basically linear in its literal sense: within a line. Instead of one index, we would have four levels, if the main character is counted twice, as for capital and small letters. This is neither well printable nor aesthetically pleasing. Au contraire, I regularly write $\exp(i \pi)$ instead of $e^{i \pi}$ for exactly this reason. And long algebraic expressions on top of a triangle is nothing less than ridiculous.
• The current notation clearly states the nature of the functions. They are by no means symmetric as the triangle suggests.
• Context-dependent notation is per se a no-go, aka to be avoided, aka a catastrophe.
• The entire discussion only makes sense in $\mathbb{R}$. However, all of them are used in hundreds of more contexts.
• The whole issue is simply a work therapy for didactic professors who ran out of ideas to reinvent the wheel for the hundredth time. I've seen so many nonsense in school books so far, this would only be another one. And its only purpose is to confuse people. There is not the least insight in it.
• Instead of seriously discussing this, we should change the title of the thread to "Triangle of Confusion and other strange ideas", move it into general discussion, as it has nothing to do with mathematics, and encourage members to invent other nonsense notations. I will start:

$$
\int f(x)\, dx = {C+}^{-1}\left( \dfrac{df}{dx} \right) \text{ and } \int_a^b f(x)\, dx = {}^{-1}{\left( \dfrac{df}{dx} \right)}_a^b \text{ or } \int f(x)\,dx = \square f\\
\lim_{n \to \infty}a_n = \overline{a_n}| \text{ and } \lim_{n \to -\infty}a_n = |\overline{a_n} \text{ and } \lim_{x \to a}f(x) = f(\hat{a})
$$
and before I will forget it: no more slashes as divisions anymore. I request a consequent use of $a^{-1}$ for $\frac{1}{a}$. This would at least make sense and solve many of the problems students have to divide and / or add fractions! But I admit it isn't fancy, just logical.

 

[Mark44]

Instead of seriously discussing this, we should change the title of the thread to "Triangle of Confusion and other strange ideas", move it into general discussion, as it has nothing to do with mathematics, and encourage members to invent other nonsense notations.

The whole issue is simply a work therapy for didactic professors who ran out of ideas to reinvent the wheel for the hundredth time. I've seen so many nonsense in school books so far, this would only be another one.

Amen!

 

[jedishrfu]

With respect the use of $\square$ I think that’s taken as the Delambertian used in GR as 4th dim analogue of the $\nabla$ operator.

https://en.m.wikipedia.org/wiki/D'Alembert_operator

 

[fresh_42]

With respect the use of $\square$ I think that’s taken as the Delambertian used in GR as 4th dim analogue of the $\nabla$ operator.

https://en.m.wikipedia.org/wiki/D'Alembert_operator

... whilst the exact same formula doesn't make use of $\triangle$?

 

[robphy]

Don't confuse this with the Ohm's Law triangle.

http://ohmlaw.com/voltage-current-resistance-triangle-vir-triangle/

And electrical "power" is $VI$, $I^2R$, and $\frac{V^2}{R}$

 

[fresh_42]

Don't confuse this with the Ohm's Law triangle.

http://ohmlaw.com/voltage-current-resistance-triangle-vir-triangle/
View attachment 232494
And electrical "power" is $VI$, $I^2R$, and $\frac{V^2}{R}$

I always used Switzerland as mnemonic: We abbreviate voltage by "U" here (don't ask me why), and then $U=RI$ becomes a Swiss canton.

 

[OmCheeto]

and encourage members to invent other nonsense notations

Ok then.
I guess it's not quite the right time to introduce my new "Diamond of Doom" idea.

 

[YoungPhysicist]

Ok then.
I guess it's not quite the right time to introduce my new "Diamond of Doom" idea.

Wow.I got interested,what exactly is that?

 

[jedishrfu]

You can define it in terms of the $\Delta$ and the $\nabla$ to magnify the triangles of power.

 

[jedishrfu]

... whilst the exact same formula doesn't make use of $\triangle$?

?

 

[fresh_42]

?

 

[jedishrfu]

View attachment 232504

Where's the unbroken circle?

Spoiler: An Unbroken Circle

 

[jedishrfu]

Well now that we have defeated the Triangle of Power, I think it time to close this thread before Thanos comes to claim it or the diamond of doom.

https://en.wikipedia.org/wiki/Thanos

Thank you all for contributing here.

 

[fresh_42]

The diamond of doom: ${}_{-1}\stackrel{\pi}{\stackrel{\Delta}{\nabla}_e}$

 

[jedishrfu]

The diamond of doom: ${}_{-1}\stackrel{\pi}{\stackrel{\Delta}{\nabla}_e}$

Oh no its the all seeing $\pi$

Ok @fresh_42 you had to make me reopen the thread...

I'll raise you one $\prod$

 

[YoungPhysicist]

I think it time to close this thread

You aren’t serious right?

 

[fresh_42]

Oh no its the all seeing $\pi$
I'll raise you one $\prod$

I assume in memoriam, Cledus. But Bandit's name is written with a $P$.

 

[jedishrfu]

Some Smoky stuff:

http://mentalfloss.com/article/80484/13-fast-facts-about-smokey-and-bandit

I couldn't find a reference for your quote but it sounds so Smoky.

 

[OmCheeto]

Wow.I got interested,what exactly is that?

It's something I made up, because the "oh-plus" function was confusing me. (Still is actually. As I said, it's going to take me a while to learn this.)

\begin{matrix}
x & & y & & x⊕y\\
o∆8 & * & o∆8 & = & o∆8\\
\\
diamond & of & doom?\\
x & & y & & x⊕y\\
o◊8 & * & o◊8 & = & o◊8\\
1/x & & 1/y & & 1/(1/x+1/y)
\end{matrix}

I think it might come in handy when first introducing the concept. It strikes me as kind of heuristic. And then as the student become familiar with the process, it morphs into the triangle.
Much like when introducing multiplication:
4 times 5 is equal to 5 + 5 + 5 + 5
which eventually morphs into
4 x 5

It becomes a "generally" understood shorthand concept.

And the diamond actually contains both the symbols for exponentiation(^) and root(√), which the 2nd grade teacher can doodle in full color. Mostly in preparation for students going on, and interacting with old teachers, who still stubbornly use the old symbols.

Btw, I researched the origins of the triangle the best I could, and found the original stackexchange thread:

2011.03.31 (7½ years ago)
https://math.stackexchange.com/questions/30046/alternative-notation-for-exponents-logs-and-roots​

Grant published his video 2⅓ years ago.

So it should probably give the old timers some solace, that only two people have looked into it after 7½ years.
(Guessing you and I make it 4 now. Though I'm not sure I'm going to like it when I get done. But I'm pretty sure I'll have learned a lot of interesting things on my way there.)

 

[OmCheeto]

But I'm pretty sure I'll have learned a lot of interesting things on my way there.

Because of this thread, somewhere in the last 12 hours, I found out that exponential and polynomial growth are two different things.*

And before anyone yells at me, I had someone on on the internets ask me the other day if I was being purposely obtuse.
I told them quite honestly; "No, I really am this stupid".

------------
*Well, they both have exponents, so they must be the same thing.
Yay learning!

 

[jedishrfu]

Is it acute obtuseness or equilaterally spread out?